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On partitions of discrete boxes

2002
*
Discrete Mathematics
*

We prove that any

doi:10.1016/s0012-365x(02)00428-4
fatcat:gsi3xpnyhnfthcf5lqhqcn5spq
*partition**of*an n-dimensional*discrete**box*into nontrivial sub-*boxes*must consist*of*at least 2 n sub-*boxes*, and consider some extensions*of*this theorem. ... We are also grateful to Jeff Kahn for suggesting the point*of*view taken in subsection 2.4. ... Theorem 1 Let A be an n-dimensional*discrete**box*, and let {B 1 , B 2 , . . . , B m } be a*partition**of*A into nontrivial sub-*boxes*. Then m ≥ 2 n . Proof. ...##
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On Partitions of Two-Dimensional Discrete Boxes
[article]

2018
*
arXiv
*
pre-print

Let A and B be finite sets and consider a

arXiv:1812.08396v1
fatcat:eblbosvmcvdz3j765y5g6kmgym
*partition**of*the*discrete**box*A × B into sub-*boxes**of*the form A' × B' where A' ⊂ A and B' ⊂ B. ... We show that a*partition**of*A × B that has the (k, ℓ)-piercing property must consist*of*at least (k-1)+(ℓ-1)+ 2√((k-1)(ℓ-1)) sub-*boxes*. ... Acknowledgments We thank Ron Holzman for pointing out a mistake in an earlier version*of*this paper. ...##
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On the number of minimal partitions of a box into boxes

2006
*
Discrete Mathematics
*

The number

doi:10.1016/j.disc.2005.10.027
fatcat:7bseg7ttpzccvjbjr6paxqmjcy
*of*minimal*partitions**of*a*box*into proper*boxes*is examined. ... At the present stage*of*investigations this problem seems to be out*of*reach, as we know little*on*the structure*of*the minimal*partitions*. ... to F satisfies*one**of*the following two relations: Lemma 6. ...##
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Performing large full-wave simulations by means of a parallel MLFMA implementation

2013
*
2013 IEEE Antennas and Propagation Society International Symposium (APSURSI)
*

The data structures

doi:10.1109/aps.2013.6711598
fatcat:uhznxf3oznflvm6kitx2ec7eie
*of*the MLFMAtree are*partitioned*according to the so-called hierarchical*partitioning*scheme, while the radiation patterns are*partitioned*in a blockwise way. ... To test the implementation*of*the algorithm, a full-wave simulation*of*a canonical example with more than 50 millions*of*unknowns has been performed. ... This method*discretizes*the scattering problem into a set*of*N linear equations, with N the number*of*unknowns*on*the*discretized*surface(s)*of*the scatterer(s). ...##
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Hierarchical parallelization of the multilevel fast multipole algorithm (MLFMA)

2013
*
Proceedings of the IEEE
*

This paper presents the hierarchical

doi:10.1109/jproc.2012.2222331
fatcat:wvgih5zlmbdlhenmccrvu4mywa
*partitioning*strategy, which provides a very efficient parallelization*of*MLFMA*on*distributed-memory architectures. ... We discuss the advantages*of*the hierarchical strategy over previous approaches and demonstrate the improved efficiency*on*scattering problems*discretized*with millions*of*unknowns. ...*One**of*the major advances in the parallelization*of*MLFMA is the development*of*the hierarchical strategy, which is based*on*the simultaneous*partitioning**of**boxes*and field samples among processors at ...##
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Range-Box Plotting Relating to Discrete Distribution

2018
*
MATEMATIKA
*

The new modication

doi:10.11113/matematika.v34.n2.935
fatcat:wdmyh6jjtbdajlpim2rgzswmu4
*of**box*plot is not based*on*the asymmetrical shape*of*distribution but more*on*the spread*of*the data and*partitioning*data into range measure. ... The existing*box*plot is constructed using ve values*of*statistics calculated from either the*discrete*or continous data. ... Table 3 shows that the standard classical*box*plot will not look at the*partitional**of*data, but really*on*the existance*of*outliers. ...##
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Simulating Microbial Community Patterning Using Biocellion
[chapter]

2014
*
Msphere
*

Biocellion further accelerates large scale, high resolution simulations using cluster computers by

doi:10.1007/978-1-4939-0554-6_16
pmid:24838890
fatcat:kmmuib5y6van5d4cgrgckh3efq
*partitioning*the work to run*on*multiple compute nodes. ... Biocellion is a highperformance software framework for accelerating*discrete*agent-based simulation*of*biological systems with millions to trillions*of*cells. ... The original yeast patterning model [5]*partitions*the simulation domain*on*top*of*the agarose cylinder into a set*of*fixed size cubic*boxes*(a*box*width is 5 μm), and a cell takes a single*box*. ...##
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The maximum discrete surface-to-volume ratio of space-filling curve partitions
[article]

2021
*
arXiv
*
pre-print

This framework thus has the potential to guide the design

arXiv:2106.12856v1
fatcat:vef5pujombhebbmjbmt4fid46i
*of*better load balancing algorithms*on*the long term. ... We investigate the surface-to-volume ratio*of*space-filling curve*partitions*for adaptive grids and derive the maximum surface-to-volume ratio as a function*of*the number*of*cells in the*partition*. ... A fundamental goal in this project is the effective parallelisation*of*a dynamically adaptive mesh refinement solver for wave questions which uses the Peano space-filling curve. ...##
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Biocellion: accelerating computer simulation of multicellular biological system models

2014
*
Bioinformatics
*

Availability and implementation: Biocellion runs

doi:10.1093/bioinformatics/btu498
pmid:25064572
pmcid:PMC4609016
fatcat:j4zdhyz7vfcqzn6az6h3zayfse
*on*x86 compatible systems with the 64 bit Linux operating system and is freely available for academic use. ...*Discrete*agentbased modeling (in combination with high-resolution grids to model the extracellular environment) is a popular approach for building biological system models. ... The extracellular space is*partitioned*into a set*of*cubical grid*boxes*. ...##
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Enabling Space Filling Curves Parallel Mesh Partitioning In Alya

2017
*
Zenodo
*

In this paper we focus

doi:10.5281/zenodo.826103
fatcat:zsub22ti7jgffg2gztcziaguwu
*on*the mesh*partitioning*, presenting the work carried out in the context*of*a PRACE Preparatory Access Project to enable a Space Filling Curve (SFC) based partitioner in the computational ... In particular, we have run our tests*on*the MareNostrum III supercomputer*of*the Barcelona Supercomputing Center. ... The sequential and parallel executions*of*the algorithm produce an almost identical*partition*if the grid used to*discretize*the bounding*box*is the same. ...##
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Optimistic planning for continuous-action deterministic systems

2013
*
2013 IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning (ADPRL)
*

Experiments

doi:10.1109/adprl.2013.6614991
dblp:conf/adprl/BusoniuDMB13
fatcat:em4v6xs5zjcwtbs7oc6hub4wsq
*on*three problems show SOOP reliably ranks among the best algorithms, fully dominating competing methods when the problem requires both long horizons and fine*discretization*. ... SOOP can be used parameter-free at the cost*of*more model calls, but we also propose a more practical variant tuned by a parameter α, which balances finer*discretization*with longer planning horizons. ... The hierarchical*partition*splits at each depth d the considered*box*along dimension k = d, into M subboxes:*one*for each*discrete*action. So, for OPD the depth in the tree is equal to the time step. ...##
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FAUST $^{\mathsf 2}$ : Formal Abstractions of Uncountable-STate STochastic Processes
[chapter]

2015
*
Lecture Notes in Computer Science
*

FAUST 2 allows refining the outcomes

doi:10.1007/978-3-662-46681-0_23
fatcat:qqltkzrnkvhdvkq6m7ld7ur6dq
*of*the verification procedures over the concrete dtMP in view*of*the quantified and tunable error, which depends*on*the dtMP dynamics and*on*the given formula. ... FAUST 2 is a software tool that generates formal abstractions*of*(possibly non-deterministic)*discrete*-time Markov processes (dtMP) defined over uncountable (continuous) state spaces. ... Left: obtained uniform*partition**of*the safe set, along with optimal safety probability for each*partition*set (color bar*on*the right). ...##
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Sylvester's bijection between strict and odd partitions

2004
*
Discrete Mathematics
*

We show that Sylvester's bijection between strict

doi:10.1016/j.disc.2002.02.001
fatcat:gkjbf2gs7rhedlvfbcgdnzj3ry
*partitions*and odd*ones*can be obtained by an appropriate coding*of**partitions*. ... The contents*of*the*boxes*may be considered as materializing a ribbon made*of*two parts, the top*one*being described by its vertical steps, the bottom*one*, by its horizontal steps, each number being the ... As a by-product,*one*obtains a statistics*on*odd*partitions*which generalizes the restriction*of*an identity*of*Bousquet-Melou and Eriksson to strict*partitions*(cf. [3] ). ...##
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Fast and accurate solutions of extremely large scattering problems involving three-dimensional canonical and complicated objects

2009
*
2009 Computational Electromagnetics International Workshop
*

We present fast and accurate solutions

doi:10.1109/cem.2009.5228112
fatcat:4iiqsuesvrgofkdbpn74g4vodu
*of*extremely large scattering problems involving three-dimensional metallic objects*discretized*with hundreds*of*millions*of*unknowns. ... Various examples involving canonical and complicated objects are presented in order to demonstrate the feasibility*of*accurately solving large-scale problems*on*relatively inexpensive computing platforms ... At this frequency, the size*of*the*box*corresponds to 875λ, and its*discretization*with the RWG functions*on*λ/10 triangles leads to matrix equations involving 174,489,600 unknowns. ...##
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How many is different? Answer from ideal Bose gas

2014
*
Journal of Physics, Conference Series
*

We show that the isobar

doi:10.1088/1742-6596/490/1/012018
fatcat:jqxnno5kz5dztjvwjnq4t7amvm
*of*the ideal gas zigzags*on*the temperature-volume plane featuring a 'boiling-like'*discrete*phase transition, provided the number*of*particles is equal to or greater than a particular ... We revisit with care the ideal Bose gas confined in a cubic*box*which is discussed in most statistical physics textbooks. ... This work was supported by the Sogang University Research Grant*of*201310030.01. ...
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